On GMW designs and a conjecture of Assmus and Key Thomas E. Norwood and Qing Xiang Dept. of Mathematics, California Institute of Technology, Pasadena,

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1 On GMW designs and a conjecture of Assmus and Key Thomas E. Norwood and Qing iang Dept. of Mathematics, California Institute of Technology, Pasadena, CA June 24, 1998 Abstract We show that a family of cyclic Hadamard designs dened from regular ovals is a sub-family of a class of dierence set designs due to Gordon, Mills and Welch [GMW]. Using a result of Scholtz and Welch [SW] on the linear span of GMW seuences, we give a short proof of a conjecture of Assmus and Key on the 2-rank of this family of designs. key words: Codes, Cyclic Dierence Sets, GMW Designs, Hadamard Designs. tnorwood@alumni.caltech.edu, xiang@cco.caltech.edu 0

2 1 Introduction We assume that the reader is familiar with the theory of designs and codes as can be found in [AK1]. A symmetric (v; k; ) design is called a Hadamard design if v = 4n? 1, k = 2n? 1, = n? 1 or v = 4n? 1, k = 2n, and = n. We rst give the denitions of two families of cyclic Hadamard designs. Unless otherwise stated, we will maintain the following notation throughout this paper: = 2 m, F is the nite eld of elements, F is the multiplicative group of F, F n denotes the n-dimensional vector space over F formed by all n-tuples with entries in F. We dene a family D() of Hadamard designs as follows. The points of D() are the nonzero elements of F. The blocks are associated with the linear functionals L : F! F 2 so that 2 B L if and only if L() = 1. Note that D() is the symmetric design developed from C() = fx 2 F jt r 2(x) = 1g, T r 2 is the trace from F to F 2. Another family of Hadamard designs of order 2 =4 = 2 2m?2 which we will call M( 2 ) may be dened by the following geometric construction. Let be a Desarguesian projective plane of order, that is, the symmetric ( ; + 1; 1) design of one- and two-dimensional subspaces of F 3. Let O consist of the points f(1; t; t 2 )jt 2 F g [ f(0; 1; 0); (0; 0; 1)g. O is a regular oval in, that is, a set of + 2 points, no three of which are collinear. Lines of that meet O in zero or two points are called exterior or secant respectively. Points not on O are called exterior points. The points of M( 2 ) are the exterior points. For each exterior point P, we dene a block B P by B P = fq 6= P jthe line through P and Q is an exterior lineg. It is easily seen that M( 2 ) is a symmetric ( 2? 1; 2 =2; 2 =4) design. It has the same parameters as D( 2 ). As all conics in are projectively euivalent, M( 2 ) is determined up to isomorphism by the parameter 2. It is known that P SL 2 () acts as an automorphism group of M( 2 ), and M( 2 ) admits a cyclic regular automorphism group (see [J]). The family M( 2 ) has been studied extensively by geometers; see especially the recent work of Maschietti [M1], [M2], [M3], and [M4]. Here we are interested in the code C 2 (M( 2 )) of M( 2 ). It is conjectured by Assmus and Key ([AK1], page 292), [AK2] that the 2-rank of M( 2 ) is m2 m?1. The purpose of this note is to give a short proof of this conjecture. 2 Main Results In this section, we rst show that M( 2 ) is a sub-family of GMW designs (which will be dened later). We will show that the conjecture of Assmus and Key follows from the result on the linear span of GMW seuences. The following construction of cyclic Hadamard designs is given by Jackson in [J]. 1

3 CONSTRUCTION 1 (Jackson [J]): Let M be a matrix with entries in F whose rows and columns are indexed by the elements of F 2 n (0; 0), with the entry M xy = xy T, x; y 2 F 2 n (0; 0), and y T is the transpose of y. The rows of M correspond to the nonzero linear functionals from F 2 to F. If f : F! f0; 1g is a map, we dene f(m) to be the matrix with (f(m)) xy = f(xy T ). Let D be a (? 1; =2; =4) cyclic dierence set in F. Dene f : F! f0; 1g by f(x) = 1 if and only if x 2 D. Then f(m) is the incidence matrix of a ( 2? 1; 2 =2; 2 =4) symmetric design. Moreover, if D 1 and D 2 are two (? 1; =2; =4) dierence sets in F, then the two designs constructed from D 1 and D 2 are isomorphic if and only if D 1 is a translate of D 2. We remark that every design obtained through the above construction is actually a cyclic dierence set. It is shown in [J] that D( 2 ) arises from Construction 1 with D = C(), and M( 2 ) arises from Construction 1 with D = 1=C() = f1=xjx 2 C()g: Next we describe a construction of cyclic dierence sets by Gordon, Mills, and Welch. In the remainder of this paper, we select and x a primitive element of F 2. Also we x = +1 as a primitive element of F. We dene a polynomial (x) as follows. Let T r 2 : F 2! F be the trace from F 2 to F. Then c i = (x) = 2?2 ( 1; if T r 2 ( i ) = 1, c i x i ; (2:1) The following is then a special case of the construction of Gordon, Mills, and Welch. CONSTRUCTION 2 (Gordon, Mills, Welch [GMW]): Let D be a (? 1; =2; =4) cyclic dierence set in F = hi and let (y) be its Hall polynomial, y = x +1. Then (x)(y) (mod x 2?1? 1) is the Hall polynomial of a ( 2? 1; 2 =2; 2 =4) cyclic dierence set in F 2 = hi. Moreover, if D 1 and D 2 are two (? 1; =2; =4) dierence sets in F, then the two dierence sets constructed from D 1 and D 2 are euivalent if and only if D 1 is a translate of D 2. (In this construction, by Hall polynomial of D, we mean that if D = f d 1 ; d 2 ; ; d =2 g F = hi, then (y) = y d 1 + y d y d =2. Similarly, if E = f e 1 ; e 2 ; ; e 2 =2 g F = hi is a ( 2? 1; 2 =2; 2 =4) dierence set in F, then its 2 2 Hall polynomial is dened to be x e 1 + x e x e 2 =2. We note that the denition of Hall polynomial here depends on the choice of primitive elements of the nite elds involved). In particular, we remark that (x)(y s ) (mod x 2?1? 1) is the Hall polynomial of a cyclic dierence set whenever g:c:d(s;? 1) = 1. The dierence sets arising from Construction 2 are called GMW dierence sets, and the corresponding designs are called GMW designs. We will now prove that Constructions 1 and 2 are essentially the same. The present proof is 2

4 adopted from the one appearing in [N]. However, we have learned that this was independently proven in [JW]. It will suce to determine the Hall polynomial of a dierence set arising from Construction 1 and show that it has the form described in Construction 2. Let M be as dened in Construction 1. Now let the columns of M be ordered as 1; ; 2 ; ; 2?2, is the xed primitive root of F 2. The rows of M may then be ordered so that M is circulant. Let r be the row of M associated with T r 2. Let x be the vector of the rst + 1 entries in r. Then r = (x; x; 2 x; ;?2 x). Let f : F! f0; 1g with f(0) = 0. To nd the Hall polynomial of any dierence set whose incidence matrix is of the form f(m) (i.e. those arising from Construction 1), we must simply nd the polynomial associated with f(r). To this end, we dene: Also We then have the following b i = (y) =?2 a i = 1; if f( i ) = 1, (x) = 2?2 ( 1; if f(t r 2 a i y i ; (2:2) b i x i ; (2:3) ( i )) = 1, Theorem 1. With notation as dened above, (x) (x)(y) (mod x 2?1? 1), y = x +1. Proof: It suces to prove the result in the case (y) is a monomial. Assume that (y) = y k, 0 k? 2, i.e. f( i ) = 1; 0 i? 2 if and only if i = k, then But if T r 2 ( i ) = 1, then T r 2 (x)(y) = 2?2 c i y k x i = 2?2 c i x k(+1)+i : ( k(+1)+i ) = T r 2 ( k i ) = k. Thus (x)(y) = d i = 2?2 d i x i ; (mod x 2?1? 1); ( 1; if T r 2 ( i ) = k, 3

5 The result now follows. This completes the proof. 2 Thus we have shown that the two constructions in this section are essentially the same. It follows that M( 2 ) is a sub-family of GMW designs since M( 2 ) arises from Construction 1 with D = 1=C(). M( 2 ) is a cyclic design, every block of M( 2 ) is a cyclic dierence set in F 2 = hi. Maintaining previous notation, we have Corollary 1. The Hall polynomial of the block of M( 2 ) corresponding to T r 2 is (x) 0 (y?1 ) (mod x 2?1? 1), 0 (y) is the Hall polynomial of C(). For brevity, we will call the dierence set in Corollary 1 D M ( 2 ). It is now easily seen that the characteristic function of the dierence set D M ( 2 ) in F = hi is T r 2[(T r 2 2 ( i ))?2 ], 0 i 2? 2. In general, the characteristic function of the dierence set in F = hi with 2 Hall polynomial (x) 0 (y s ) (mod x 2?1? 1) is T r2[(t r 2 ( i )) r ], 0 i 2? 2, rs 1 (mod? 1). The seuences ft r2[(t r 2 ( i )) r ]g, 0 i 2? 2, g:c:d(r;? 1) = 1, are called GMW seuences which were studied in [SW], [AB]. We now consider the 2-rank of the design M( 2 ), i.e. the F 2 -dimension of the vector space spanned by the rows (the blocks) of the incidence matrix of M( 2 ). Since M( 2 ) is a cyclic design, the 2-rank of M( 2 ) is just the F 2 -dimension of the ideal generated by the Hall polynomial of M( 2 ) in F 2 [x]=(x 2?1? 1). It turns out that this was studied by several engineers [SW], [AB] in terms of linear span (or linear complexity) of GMW seuences. The linear span of a periodic seuence fa i g of elements from F with period v (here is a power of any prime) is the minimal degree of a linear feedback shift register (LFSR) for generating fa i g. It can be shown ([AK1], page 69) that the linear span of a periodic seuence fa i g of elements from F with period v is just the F -dimension of the ideal generated by Pv?1 a i x i in F [x]=(x v?1). Therefore, by our discussion following Corollary 1, we see that the 2-rank of M( 2 ) is just the linear span of the periodic seuence fb i g, b (i mod 2?1) = T r2[(t r 2 ( i ))?2 ], 0 i 2? 2. We now uote the following theorem on the linear span of GMW seuences from [SW]. For more general formulas of the linear span of p-ary GMW seuences, we refer the reader to [AB]. In the following theorem, F 2 M is the nite eld of 2 M elements, M = JK, T r 2M 2 J, T r 2J 2 are the trace from F 2 M to F 2 J and the trace from F 2 J to F 2 respectively. Theorem A. Let fa i g be a GMW seuence whose elements are given by a i = T r 2J 2 [(T r 2M 2 J (i )) r ] 2 F 2 ; is a primitive element of F 2 M and r; 0 < r < 2 J? 1, is relatively prime to 2 J? 1. Then the linear span L of fa i g is given by L = J(M=J) w ; 4

6 w is the number of ones in the base-2 representation of r. We remark that the main idea in the proof of Theorem A is the use of Fourier transform. As a corollary of Theorem A, we have Corollary 2. The 2-rank of M( 2 ) is m2 m?1. Proof: In Theorem A, let M = 2m, J = m, r =? 2 = m?1. Then w = m? 1. The 2-rank of M( 2 ) euals to the linear span of fb i g, which by Theorem A, is m2 m?1. This completes the proof. 2 We remark that the seuence associated with M( 2 ) has the maximal linear span among the GMW seuences ft r2[(t r 2 ( i )) r ]g, g:c:d(r;? 1) = 1. We also mention that many other properties of the code C 2 (M( 2 )) were studied in [N]. ACKNOWLEDGMENT The authors thank the anonymous referee for his/her careful reading of the paper. 5

7 References [AB] [AK1] [AK2] M. Antweiler and L. Bomer, Complex seuences over GF (p m ) with a two-level autocorrelation function and a large linear span, IEEE Inform. Theory, Vol.38, No.1 (1992) E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge (1992). E. F. Assmus, Jr. and J. D. Key, Hadamard matrices and their designs: a coding theoretic approach, Trans. Amer. Math. Soc., 330 (1992), [GMW] B. Gordon, W. H. Mills, and L. R. Welch, Some new dierence sets, Can. J. Math., 14 (1962), [J] [JW] Wen-Ai Jackson, A characterization of Hadamard designs with SL(2; ) acting transitively, Geom. Dedicata, 46 (1993), Wen-Ai Jackson and Peter R. Wild, On GMW designs and cyclic Hadamard designs, preprint. [M1] A. Maschietti, Hyperovals and Hadamard designs, J. Geometry, 44 (1992), [M2] A. Maschietti, On 2 =4-sets of type (0; =4; =2) in projective planes of order 0 (mod 4), Disc. Math., 129 (1994), [M3] A. Maschietti, Regular triples with respect to a hyperoval, Ars Comb., 39 (1995), [M4] [N] [SW] A. Maschietti, On Hadamard designs associated with a hyperoval, J. Geometry, 53 (1995), T. E. Norwood, Codes and polynomials in the study of cyclic dierence sets, Ph. D. Thesis, California Institute of Technology, May, R. A. Scholtz and L. R. Welch, GMW seuences, IEEE Trans. Inform. Theory, Vol. 30, No.3 (1984),

Singer and GMW constructions (or generalized GMW constructions), little else is known about p-ary two-level autocorrelation sequences. Recently, a few

Singer and GMW constructions (or generalized GMW constructions), little else is known about p-ary two-level autocorrelation sequences. Recently, a few New Families of Ideal -level Autocorrelation Ternary Sequences From Second Order DHT Michael Ludkovski 1 and Guang Gong Department of Electrical and Computer Engineering University of Waterloo Waterloo,